$\dfrac{ 5e - 7f }{ 10 } = \dfrac{ 9e + 4g }{ -9 }$ Solve for $e$.
Solution: Multiply both sides by the left denominator. $\dfrac{ 5e - 7f }{ {10} } = \dfrac{ 9e + 4g }{ -9 }$ ${10} \cdot \dfrac{ 5e - 7f }{ {10} } = {10} \cdot \dfrac{ 9e + 4g }{ -9 }$ $5e - 7f = {10} \cdot \dfrac { 9e + 4g }{ -9 }$ Multiply both sides by the right denominator. $5e - 7f = 10 \cdot \dfrac{ 9e + 4g }{ -{9} }$ $-{9} \cdot \left( 5e - 7f \right) = -{9} \cdot 10 \cdot \dfrac{ 9e + 4g }{ -{9} }$ $-{9} \cdot \left( 5e - 7f \right) = 10 \cdot \left( 9e + 4g \right)$ Distribute both sides $-{9} \cdot \left( 5e - 7f \right) = {10} \cdot \left( 9e + 4g \right)$ $-{45}e + {63}f = {90}e + {40}g$ Combine $e$ terms on the left. $-{45e} + 63f = {90e} + 40g$ $-{135e} + 63f = 40g$ Move the $f$ term to the right. $-135e + {63f} = 40g$ $-135e = 40g - {63f}$ Isolate $e$ by dividing both sides by its coefficient. $-{135}e = 40g - 63f$ $e = \dfrac{ 40g - 63f }{ -{135} }$ Swap signs so the denominator isn't negative. $e = \dfrac{ -{40}g + {63}f }{ {135} }$